# Questions tagged [cv.complex-variables]

Complex analysis, holomorphic functions, automorphic group actions and forms, pseudoconvexity, complex geometry, analytic spaces, analytic sheaves.

2,073
questions

**4**

votes

**1**answer

46 views

### Riemann $P$-symbol for ODEs

Good afternoon, everyone. Does anyone know what the Riemann $P$-symbols mean when they contain more than three columns (i.e., to what ordinary differential equations they correspond)?
Examples: $P\...

**0**

votes

**1**answer

123 views

### Books on complex analysis for self learning that includes the Riemann zeta function?

I am searching for an introductory book in the field of complex analysis for self learning, that would contain the following:
Analytic number theory : the connection between complex analysis and
...

**10**

votes

**1**answer

329 views

### turn $\pi/n$, move $1/n$ forward

start at the origin, first step number is 1.
turn $\pi/n$
move $1/n$ units forward
Angles are cumulative, so this procedure is equivalent (finitely)
to
$$
u(k):=\sum_{n=1}^{k} \frac{\exp(\pi i H_{n}...

**8**

votes

**1**answer

304 views

### Collinear Galois conjugates

This is inspired by this old question, which may provide a bit more background. But the two present questions seem somewhat more fundamental to me.
Let $p$ be an irreducible polynomial with integer ...

**0**

votes

**1**answer

53 views

### On the integral $I_s = \int_{1}^{\infty} (\pi(x)-Li(x))x^{-s-1} \mathrm{d}x$-follow up question

This is a follow up on On the integral $I_s =\int_{1}^{\infty} (\pi(x)-Li(x))x^{-s-1} dx$
According to the answer that i got, $I_s$ is not known to converge for any real $s<1$. But suppose $I_s$ ...

**2**

votes

**2**answers

140 views

### On the integral $I_s =\int_{1}^{\infty} (\pi(x)-Li(x))x^{-s-1} dx$

Define $\pi(x)$ to be the prime counting function and Li(x) the logarithmic integral. Let $I_s$ be defined as above.
Is $I_s$ known to be convergent for any real number $s<1$ ?

**6**

votes

**0**answers

282 views

### Cheap bound on $\zeta'(s)/\zeta(s)$ or $L'(s,\chi)/L(s,\chi)$?

Say you are proving an explicit formula for $L(s,\chi)$ and/or the prime number theorem (in arithmetic progressions or not) in the usual way -- that is, shifting a line of integration from $\Re(s) = 1^...

**3**

votes

**1**answer

71 views

### Singular set of entire functions

Recently I've started studying the theory of singular values for entire functions so I'm far from being a specialist in this field. In the literature I came across the following results:
In [1] Gross ...

**1**

vote

**0**answers

133 views

### Holomorphic functions to complex torus

Let $X$ be a complex algebraic variety and $T$ a complex torus (not necessarily algebraic).
Assume that $X$ is a proper subset of its completion $\bar{X}$.
Let $f:X \to T$ be a holomorphic map.
Are ...

**4**

votes

**0**answers

31 views

### A relaxed form of quantitative transversality

Let $f$ be a holomorphic function on the unit ball $B_1(0) \subseteq \mathbb{C}^n$. Then given any pair of constants $\delta, \epsilon > 0$, does there exist a smooth function $g: B_1(0) \to \...

**5**

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227 views

### How strange are Hölder domains?

Let $D$ be a Jordan domain. We assume that $D$ is a H?lder domain. Namely, there exists a H?lder continuous Riemann map $f$ such that $D=f(\mathbb{D})$, where $\mathbb{D}$ is an open unit disk.
It ...

**2**

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**0**answers

94 views

### How to calculate the contour integration?

I encountered some problems when I read the paper. The details are shown as follows.
\begin{equation}
pdf(I)=e^I \prod_{i=1}^n \left(-j a_i\right)\int \frac{dk}{2\pi}\frac{e^{jk\left(e^I-1\right)}}{\...

**13**

votes

**1**answer

295 views

### Does the $\overline{\partial}$ operator have closed image?

Let $X$ be a complex-analytic manifold, not necessarily compact.
Does $\overline{\partial} : C^\infty(X) \rightarrow \Omega^{0,1}(X)$ have closed image with respect to the Fréchet topology given by ...

**1**

vote

**1**answer

60 views

### Disk with punctures and convex geodesical hull of the punctures isomorphic?

Consider a unit disk with marked points $z_i$, $i=1, \dots , n$ on its boundary.
Let us call this surface $X$.
As it is well known, the disk can be equipped with an hyperbolic metric and is then ...

**0**

votes

**0**answers

42 views

### Looking for a simple proof of approximating almost periodic functions by generalised trigonometric polynomials with restricted Bohr spectrum

Let $AP$ be the space of almost periodic functions, defined to be the uniform closure in $C_b(\mathbb R,\mathbb C)$ of the set of generalized trigonometric polynomials
$Q(t):=\sum_{j=1}^n a_j e^{i\...